Neumann eigenvalue problems on the exterior domains

Abstract

For $p\in (1,\infty )$, we consider the following weighted Neumann eigenvalue problem on $B_1^c$, the exterior of the closed unit ball in $R^N$: \begin{equation}\label{Neumann eqn} \begin{aligned} -Δ_p ϕ& = λg |ϕ|^{p-2} ϕ\ \text{in}\ B^c_1, \\ \displaystyle\frac{\partial ϕ}{\partial ν} &= 0 \ \text{on} \ \partial B_1, \end{aligned} \end{equation} where ${\Delta }_p$ is the $p$-Laplace operator and $g\in L_{loc}^1(B_1^c)$ is an indefinite weight function. Depending on the values of $p$ and the dimension $N$, we take $g$ in certain Lorentz spaces or weighted Lebesgue spaces and show that the above eigenvalue problem admits an unbounded sequence of positive eigenvalues that includes a unique principal eigenvalue. For this purpose, we establish the compact embeddings of $W^{1,p}(B_1^c)$ into $L^p(B_1^c,|g|)$ for $g$ in certain weighted Lebesgue spaces. For $N>p$, we also provide an alternate proof for the embedding of $W^{1,p}(B_1^c)$ into $L^{p^{\ast },p}(B_1^c)$. Further, we show that the set of all eigenvalues is closed.